Conjecture on tangent-map morphisms for rational curves in projective space

Let $d\geq n+1$ be integers. Set

$$q=\left\lfloor \frac{d}{n} \right\rfloor,\qquad a=(q+1)n-d,\qquad b=d-nq.$$

For a morphism

$$f:\mathcal{O}_{\mathbb{P}^1}(2)\longrightarrow \mathcal{O}_{\mathbb{P}^1}(d+q)^{\oplus a}\bigoplus \mathcal{O}_{\mathbb{P}^1}(d+q+1)^{\oplus b},$$

where $C\xrightarrow{\phi}\mathbb{P}^n$ denotes a rational curve of degree $d$, the following assertions hold. Tangent-map conjecture. (1) If $\operatorname{char}(K)=0$, $n=2,3$ or $d\geq n+2$, then a general such morphism $f$ is induced by a rational curve, namely there is a curve $C\xrightarrow{\phi}\mathbb{P}^n$ such that $f=\phi^*\circ d\phi$. (2) If $n\geq4$, there exists a morphism

$$f:\mathcal{O}_{\mathbb{P}^1}(2)\longrightarrow \mathcal{O}_{\mathbb{P}^1}(n+1)^{\oplus(n-1)}\oplus\mathcal{O}_{\mathbb{P}^1}(n+3)$$

that is not induced by any rational curve $C\xrightarrow{\phi}\mathbb{P}^n$ of degree $n+1$, so that $f\ne d\phi\circ\phi^*$ for every such curve. The question concerns which morphisms arise from composing the tangent map of a rational curve with restriction to the curve; the case $d\geq n+1$ was stated as open before these main results, while the displayed assertions provide the claimed positive and negative answers in the indicated cases.